A location-inventory-pricing model in a closed loop supply chain network with correlated demands and shortages under a periodic review system

We consider a three-level closed loop supply chain for location-inventory-pricing decisions where demands across the customer zones are correlated, the inventory policy is a periodic review (R,T), and face shortages. The problem is to determine the number and the location of the collection and distribution centers (CDCs) and the plants to be opened, the assignment of the customer zones to the CDCs and the CDCs to the plants, the price of the new product, the incentive value of the returned product, the inventory review intervals at the CDCs, and the fraction of the demand backordered during the shortage period at the CDCs such that the total profit is maximized. We propose a nonlinear programming model for this purpose. Since this problem is one of the NP-hard problems three meta-heuristic algorithms are used for solving it namely the genetic algorithm (GA), the imperialist competitive algorithm (ICA), and the firefly algorithm (FA). We present an encoding and decoding procedure to represent solutions. To attain the reliable results in the proposed algorithms, the Taguchi method is utilized for calibrating the parameters of these algorithms. The computational results show that in terms of the objective function value there is no significant difference between the mentioned algorithms but in terms of the CPU time the firefly algorithm compared to the other algorithms needs more time for solving the problem.